Institutions that are active in financial markets routinely apply mathematically based models to value financial instruments in their trading portfolios. A valuation model typically takes as input the current values of a number of market rates and gives as output a theoretical price, or fair value, of the instrument in question. This allows the valuation of a financial instrument that may not have been traded recently, provided that the input market rates are available.
These input market rates are typically the traded prices of simpler financial instruments of which an example is a foreign exchange rate. Market rates are only known if the trade took place in the relevant instrument during the time interval of interest. If a trade did not take place, one could use the rate corresponding to the last trade that took place, but that will cause inaccuracy because of the lack of currency of the rate.
A basic use of valuation models is to determine the daily profit and loss of trading portfolios. This involves comparing the value of a trading portfolio at the end of one day with its value at the end of the previous trading day, and requires the valuation of financial instruments that may not have traded in the course of the trading day. Valuation models are also used to inform trading decisions and to assess the risks arising out of trading portfolios. Market risk is the risk of the value of a trading portfolio decreasing. Credit risk is the risk of a counter-party of a trade defaulting on contractual obligations. The amount of loss attributable to a default depends, at least in part, on the market value of the trades of the defaulting counter-party at the time of default.
The analysis of market risk involves making probabilistic assumptions as to how market rates may change in the future. The impact of possible market rate changes on the value of the trading portfolio is then quantified and various measures of risk can be calculated.
A common method of market risk assessment is historical simulation. Historically observed rate changes of several rate series together are assumed to be statistically independent and to form a representative random sample and are applied to current rates. The resulting sets of rates (one for each historical time interval) are then used to revalue an existing trading portfolio. This gives a set of a hypothetical future portfolio values used to calculate measures of market risk arising out of the trading portfolio. However, this method disadvantageously requires gaps in the historical rate series to have been filled before the analysis begins. It also assumes that a complete set of current rates exists.
One known approach is to set each unknown rate to its previous known value. This can cause long sequences of repeated values, which leads to underestimation of risk, and sudden large jumps to the next known value, which leads to overestimation of risk.
Another approach is to fill gaps in each rate series by linearly interpolating between the known rates. This involves graphing, for each rate series, the known values, drawing straight lines between consecutive known values and reading off the unknown rates from the resulting graph. This approach tends to cause underestimation of risk because it ignores the inherent variation in rates and does not take into account the fact that the rate series are correlated.
A significant disadvantage of prior art methods of modeling unknown rates is that they do not take into account all known rates, including those of other rate series and those known at other times, that are related to the unknown rates. The effect of this is inaccuracy in any pricing or analysis of risk that depends on such rates.